Different expressions about time reversal and confusion

[hokhani]

TL;DR

Discrepancy between Sakurai and Bruus

In Sakurai, section 4.4, says that if $\psi(x,t)$ is a solution of Schrödinger equation then $\psi^*(x,-t)$ is always another solution. However, in the corrected version of "Many-body Quantum Theory in Condensed Matter Physics, Bruus and Flensberg, 2016", section 7.1.4, restricts this condition and says that if we have time reversal symmetry $H=H^*$ then $\psi^*(x,-t)$ would be another solution. I would like to know which of these statements is correct.

 

[PeroK]

TL;DR Summary: Discrepancy between Sakurai and Bruus

In Sakurai, section 4.4, says that if $\psi(x,t)$ is a solution of Schrödinger equation then $\psi^*(x,-t)$ is always another solution. However, in the corrected version of "
Many-body Quantum Theory in Condensed Matter Physics, Bruus and Flensberg, 2016", section 7.1.4, restricts this condition and says that if we have time reversal symmetry $H=H^*$ then $\psi^*(x,-t)$ would be another solution. I would like to know which of these statements is correct.

Sakurai must assume that the Hamiltonian has that property in general. It's not hard too see from the SDE that $H = H^*$ is required for this to make sense.

 

[hokhani]

Thanks, it seems that for the Hamiltonian in Sakurai, $H=\frac{P^2}{2m}+V(x)$, which is in the position space, the Hermitian condition $H=H^{\dagger}$ fulfils automatically $H=H^*$ since $V^\dagger=V$ demands $V=V^*$.